Factual error: Spock presents himself to Dr. McCoy in sickbay to be relieved of duty for "killing" Captain Kirk. However, as a medical officer, Bones has no place in the ship's chain of command. Doctors have an officer's rank in recognition of their education and value to the service, but no authority over military matters. (For example should Kirk, Spock and Scotty be indisposed, command of the ship would fall to Sulu who is in the chain of command even though McCoy has a higher rank) If Spock believes he should be relieved for breaking regulations and committing murder, and thus confined until court-martial, Scotty would be the one to do it, as he is the next senior officer. And this is not a character mistake, Spock quotes regulations regularly, there is no way he'd ignore chain of command. Contrary to what viewers seem to think, militaries have regulations and officers don't get to ignore them and do what they feel like! The only reason Spock talks to McCoy is so Kirk can walk out of the back of sickbay and create a nice emotional moment.
Add timeGrumpy Scot
Factual error: The Enterprise accidentally travels a short distance outside the galaxy and can't find its way back. But they'd have to travel for months to get so far outside the Milky Way that they couldn't, well, just turn around in the void and see it. Our galaxy is huge. 100,000 light years across. Very huge. And that barrier may surround the galaxy, but even it is big, pink and visible.
Add timeJean G
Character mistake: When Spock beams down after calculating there was just enough energy left for him to do so, Sulu and Kirk watch him materialize, and Sulu says, "Someone beaming down from the bridge." From the bridge? Shouldn't he have said "from the ship"? No one ever beamed down directly from the bridge, but even if they could, how would Sulu know that's where they'd come from?
Continuity mistake: Just after the ship is first attacked, Scotty tells McCoy, "We can't fire full phasers with our screens up." Say what? The Enterprise couldn't operate transporters with the shields up, but it fired its phasers, full or otherwise, with the screens up every time it went into battle, and always had. If they'd been forced to shut their shields down every time they fired, the Enterprise would have been history long before this.
Continuity mistake: When Kirk and Spock are chatting in Kirk's quarters, they finish their discussion and as they head out Kirk orders the ship to red alert. Then after the commercial break, they're on the bridge, the Gorn ship is at a dead halt, and Kirk orders the ship to red alert again. With a hostile ship in the vicinity, Kirk, a highly experienced captain, would hardly have taken the ship off red alert.
Factual error: When the Enterprise is slingshotting its way back to the "present" it uses the computer to deposit the two men at different times, The Enterprise is traveling backwards in time, so will logically meet the time when the guard was beamed up first, not after Christopher is returned to his cockpit. So the order is wrong. Should be guard then Christopher, not Christopher then guard.
Factual error: In the opening scene on the bridge, when Spock states the planet's properties, the circumference is given in US miles; the mass is given in metric tons; the density is given in metric grams per cubic centimeter; and the atmosphere is given as oxygen/nitrogen. No scientist of Spock’s standing would mix US and metric unit systems. The atmosphere composition should also be stated reversed as “nitrogen/oxygen” with the most abundant gas first.